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Thread: Vector Calculus: Flux and divergence theorem

  1. #1
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    Unhappy Vector Calculus: Flux and divergence theorem

    1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2)
    A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards.
    B)Verify Gauss's divergence theorem for the cube and the vector field F by computing each side of the formula.
    C)Using Gauss's divergence theorem evaluate the flux $\displaystyle \int \int F.dS $ of the vector field F=xi+yj+$\displaystyle z^2$k where S is a closed surface consisting of the cylinder$\displaystyle x^2 + y^2 = a^2$, 0<z<b and the circular disks $\displaystyle x^2 + y^2 , a^2$ at z=0 and $\displaystyle x^2 + y^2 = a^2$ at z=b.

    I have the basics but dont know how to get an answer. My workings are attatched.
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    Last edited by bobmarley909; May 19th 2010 at 01:28 PM. Reason: maths editing
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  2. #2
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by bobmarley909 View Post
    1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2)
    A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards.
    B)Verify Gauss's divergence theorem for the cube and the vector field F by computing each side of the formula.
    C)Using Gauss's divergence theorem evaluate the flux $\displaystyle \int \int F.dS $ of the vector field F=xi+yj+$\displaystyle z^2$k where S is a closed surface consisting of the cylinder$\displaystyle x^2 + y^2 = a^2$, 0<z<b and the circular disks $\displaystyle x^2 + y^2 , a^2$ at z=0 and $\displaystyle x^2 + y^2 = a^2$ at z=b.

    I have the basics but dont know how to get an answer. My workings are attatched.

    $\displaystyle z= 0 ,n= -k ,F ^. n = 0 $

    $\displaystyle z = 2, n= k, F^.n =0$

    $\displaystyle y = 0, n= -j, F^.n = 0$

    $\displaystyle y = 2, n=j, F^. n = 0$

    $\displaystyle x=0, n=-i, F^. n = 0$

    $\displaystyle x = 2, n = i, F^. n =2$

    So now

    $\displaystyle \int^{2}_{0} \int^{2}_{0} 2 dxdy = 8$

    Using divergence theorem

    $\displaystyle \int^{2}_{0} \int^{2}_{0}\int^{2}_{0} dxdydz = 8$

    c)

    $\displaystyle \int^{2\pi}_{0} \int^{a}_{0} \int^{b}_{0} (2 +2z)r dzdrd\theta$
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